Rigidity of Eigenvalues of Generalized Wigner Matrices

Introduction

Random matrices were introduced by E. Wigner to model the excitation spectrum of large nuclei. The central idea is based on the observation that the eigenvalue gap distribution for a large complicated system is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. As a special case of this general belief, the eigenvalue gap distribution of random matrices should be independent of the probability distributions of the ensembles and thus is given by the classical Gaussian ensembles. Besides the eigenvalue gap distribution, similar predictions hold also for short distance correlation functions of the eigenvalues. Since the gap distribution can be expressed in terms of correlation functions, mathematical analysis is usually performed on correlation functions. From now on, we refer to universality for the fact that the short distance behavior of the eigenvalue correlation functions of a random matrix ensemble are the same as those of the Gaussian ensemble of the same symmetry class (Gaussian unitary, orthogonal or symplectic ensemble, i.e., GUE, GOE, GSE).

The universality question can be roughly divided into the bulk universality in the interior of the spectrum and the edge universality near the spectral edges. Over the past two decades, spectacular progress on bulk and edge universality was made for invariant ensembles, see, e.g., and for a review. For non-invariant ensembles with i.i.d. matrix elements (Standard Wigner ensembles) edge universality can be proved via the moment method and its various generalizations, see, e.g., . In a striking contrast, the only rigorous results for the bulk universality of non-invariant Wigner ensembles were the work by Johansson and subsequent improvements on Gaussian divisible Hermitian ensembles, i.e., Hermitian ensembles of the form

where $H_{0}$ is a Wigner matrix, $V$ is an independent standard GUE matrix and $s$ is a fixed positive constant independent of $N$. The Hermitian assumption is essential since the key formula used in and the earlier work is valid only for Hermitian ensembles.

The bulk universality, however, was expected to hold for general classes of Wigner matrices, see Mehta’s book , Conjectures 1.2.1 and 1.2.2 on page 7. We will refer to these two conjectures as the Wigner-Dyson-Gaudin-Mehta conjecture due to their pioneering work. Until a few years ago this conjecture remained unsolved, mainly due to the fact that all existing methods on local eigenvalue statistics depended on explicit formulas which were not available for Wigner matrices. In a series of papers , we developed a new approach to understand local eigenvalue statistics. This approach, in particular, led to the first proof of the Wigner-Dyson-Gaudin-Mehta conjecture for Hermitian Wigner matrices with smooth distributions for the matrix elements. We now give a brief summary of this approach which motivates the current paper.

The first step was to derive a local semicircle law, a precise estimate of the local eigenvalue density, down to energy scales containing around $N^{\varepsilon}$ eigenvalues. In fact, we also obtain precise bounds on the matrix elements of the Green function. The second step is a general approach for the universality of Gaussian divisible ensembles by embedding the matrix (1.1) into a stochastic flow of matrices and use that the eigenvalues evolve according to a distinguished coupled system of stochastic differential equations, called the Dyson Brownian motion . The central idea is to estimate the time to local equilibrium for the Dyson Brownian motion with the introduction of a new stochastic flow, the local relaxation flow, which locally behaves like a Dyson Brownian motion but has a faster decay to global equilibrium. This approach entirely eliminates the usage of explicit formulas and it provides a unified proof for the universality of Gaussian divisible ensembles for all symmetry classes. Furthermore, it also gives a conceptual interpretation that the origin of the universality is due to the local ergodicity of Dyson Brownian motion.

More precisely, we will use a slightly different version of (1.1), namely

to ensure that the variance of $H_{t}$ remains independent of $t$. Denote by $\lambda_{j}$ the $j$-th eigenvalue of the random matrix $H_{t}$, labelled in increasing order, $\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{N}$, and $\gamma_{j}$ the classical location of the $j$-th eigenvalue, i.e., $\gamma_{j}$ is defined by

where $\varrho_{sc}(x)=\frac{1}{2\pi}\sqrt{(4-x^{2})_{+}}$ is the semicircle law. Our main result on the universality for the Dyson Brownian motion states that, roughly speaking, the short distance correlation functions for $H_{t}$ at the time $t\sim N^{-2{\mathfrak{a}}}$ and $H_{t=\infty}$ are identical in weak sense provided that the following main condition holds:

Assumption III. There exists an ${\mathfrak{a}}>0$ such that

Once the universality for the Gaussian divisible ensemble is established, the last step is to approximate all matrix ensembles by Gaussian divisible ones. This step can be done via a reverse heat flow argument for ensembles with smooth probability distributions or more generally via the Green function comparison theorem which compares the distributions of eigenvalues of two ensembles around a fixed energy. The key input for the latter approach was to prove a-priori estimates on the matrix elements of the Green function. These estimates have been obtained together with the local semicircle law.

To summarize, our approach to universality consists of the following three main steps: Step 1. Local semicircle law. Step 2. Universality for Gaussian divisible ensembles. Step 3. Approximation by Gaussian divisible ensembles. Both Step 2 and 3 rely on a strong local semicircle law from the Step 1.

Shortly after the preprint appeared, another method for the universality was posted by Tao and Vu . This method contains similar three ingredients as in ; their key result, prior to the Green function comparison theorem appeared in , states that the probability distributions of the $j$-th eigenvalue of two ensembles for a fixed label $j$ in the bulk are identical as $N\to\infty$ provided that the first four moments of the matrix elements of the two ensembles are identical. This result also implies the universality of the correlation functions for Hermitian Wigner ensembles by combining it with the Gaussian divisible results of for the Step 2. For symmetric ensembles, it requires the first four moments matching those of GOE. As in our approach, a key analytic input for is the local semicircle law established in . The bulk universality in the case of symmetric matrices in the generality as stated in Mehta’s book (in particular, without the assumption to match four moments), was proved in . The key input is to link universality to local ergodicity of Dyson Brownian motion, reviewed in the previous paragraphs.

Due to the fundamental role of the local semicircle law, its error estimates were improved many times since its first proof in . Furthermore, it was extended to sample covariance ensembles and generalized Wigner ensembles whose matrix elements are allowed to have different but comparable variances. The best existing error estimates for local semicircle law of generalized Wigner ensembles, given in , are already almost optimal in the bulk of the spectrum, but not near the edges. In this paper, we will prove a strong local semicircle law, Theorem 2.1, which, up to $\log N$ factors, gives optimal error estimates everywhere in the spectrum. There are four important consequences of this result:

It implies that Assumption III holds with the right hand side of (1.4) given by $N^{-2}(\log N)^{C\log\log N}$ for some constant $C$, i.e., ${\mathfrak{a}}$ can be chosen arbitrary close to 1/2. Thus the Dyson Brownian motion reaches local equilibrium at $t\sim N^{-1+\delta}$ for arbitrary small $\delta$. Up to the factor $N^{\delta}$, this is optimal. Since the time to the global equilibrium for the Dyson Brownian motion is order one, we have thus established Dyson’s conjecture that the Dyson Brownian motion reaches equilibrium in two well-separated stages with time scales of order one and $N^{-1}$. As a historical note, we mention that Dyson had obtained the two time scales via heuristic physical argument and commented that a rigorous proof of his prediction is lacking. Furthermore, the notion of local equilibrium was used by Dyson in a very vague sense, see for a more detailed discussion.

It implies certain explicit error estimates for the universality of correlation functions in short scales.

It implies the rigidity of eigenvalues in the sense that

for some positive constants $C$ and $c$. In other words, the eigenvalue is near its classical location with an error of at most $N^{-1}(\log N)^{C\log\log N}$ for generalized Wigner matrices in the bulk and the estimate deteriorates by a factor \big{(}\frac{N}{j}\big{)}^{1/3} near the edge $j\ll N$.

It implies the edge universality in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are equal in the large $N$ limit provided that the second moments of the two ensembles are identical. We recall the standard assumption that the first moments of the matrix elements are always zero for all generalized Wigner ensembles. The comparison between our edge universality theorem and the previous results will be given at the end of Section 2 after the statement of Theorem 2.4.

It is well-known that the gaps between extremal eigenvalues and their fluctuations are of order $N^{-2/3}$. Thus the edge deterioration factor in (1.5) is the natural interpolation between $N^{-1}$ in the bulk and $N^{-2/3}$ on the edges. The surprising feature of the rigidity estimate is that even if one eigenvalue is at a slightly wrong location, the probability is already extremely small. We remark that, without the $(\log N)^{C\log\log N}$ factor, the rigidity estimate (1.5) would be wrong since, at least for the classical GUE or GOE ensembles, the eigenvalues are known to fluctuate on a scale $\sqrt{\log N}/N$, see . For these ensembles, the distribution of $\lambda_{j}-\gamma_{j}$ is Gaussian in the bulk. However, the rigidity estimate (1.5) in this strong probabilistic form was not available even for the classical Gaussian ensembles.

Main results

Let $H=(h_{ij})_{i,j=1}^{N}$ be an $N\times N$ hermitian or symmetric matrix where the matrix elements $h_{ij}=\overline{h}_{ji}$, $i\leq j$, are independent random variables given by a probability measure $\nu_{ij}$ with mean zero and variance $\sigma_{ij}^{2}\geq 0$:

The distribution $\nu_{ij}$ and its variance $\sigma_{ij}^{2}$ may depend on $N$, but we omit this fact in the notation. Denote by $B:=\{\sigma^{2}_{ij}\}_{i,j=1}^{N}$ the matrix of variances. The following assumptions on $B$ are made throughout the paper:

Thus $B$ is symmetric and double stochastic and, in particular, it satisfies $-1\leq B\leq 1$.

We assume that there exists two positive constants, $\delta_{-}$ and $\delta_{+}$, independent of $N$, such that

There is a constant $C_{0}$, independent of $N$, such that

For the orientation of the reader, we mention two special cases that provided the main motivation for our work.

Example 1. Generalized Wigner matrix. Define ${C_{inf}}(N)$ and ${C_{sup}}(N)$ by

The ensemble is called generalized Wigner ensemble provided that

for some $C_{\pm}$ independent of $N$. In this case, one can easily prove that $1$ is a simple eigenvalue of $B$ and (2.3) holds with some

i.e., apart from the trivial eigenvalue, the spectrum of $B$ is separated away $\pm 1$ by positive constants that are independent of $N$. The special case ${C_{inf}}={C_{sup}}=1$ reduces to the standard Wigner matrices.

Example 2. Certain band matrices with bandwidth of order $N$. Band matrices are characterized by the property that $\sigma_{ij}^{2}$ is a function of $|i-j|$ on scale $W$, which is called the bandwidth. More precisely, the variances of a band matrix with bandwidth $1\leq W\leq N/2$ are given by

It is easy to see that many band matrices satisfy the spectral assumption (2.3). The lower spectral bound, $-1+\delta_{-}\leq B$ with some $\delta_{-}>0$ depending only on $f$, holds for any sufficiently large $W$, see Appendix A of . The parameter $\delta_{+}$ in the upper spectral bound typically behaves as of order $(W/N)^{2}$. Thus, for the condition (B) to hold, we need to assume that the bandwidth is comparable with $N$, i.e., it satisfies $W\geq cN$ with some positive constant $c$. The same assumption also guarantees that condition (C) holds.

We remark that the special case $W=N/2$ and $f(x)\geq c>0$ for $|x|\leq 1$ was already covered by Example 1, but Example 2 allows more general band matrices that may have vanishing variances. For example, with the choice of $f(x)=\frac{1}{2}\cdot{\bf 1}(|x|\leq 1)$, the ensemble with variances

The Stieltjes transform of the empirical eigenvalue distribution of $H$ is given by

Define $m_{sc}(z)$ as the unique solution of

with positive imaginary part for all $z$ with ${\mathfrak{Im}\,}z>0$, i.e.,

where the square root function is chosen with a branch cut in the segment $$so that asymptotically$\sqrt{z^{2}-4}\sim z$at infinity. This guarantees that the imaginary part of$m_{sc}$is non-negative for$\eta={\mathfrak{Im}\,}z>0$and in the$\eta\to 0$ limit it is the Wigner semicircle distribution

The Wigner semicircle law states that $m_{N}(z)\to m_{sc}(z)$ for any fixed $z$, i.e., provided that $\eta={\mathfrak{Im}\,}z>0$ is independent of $N$. Let $z=E+i\eta$ $(\eta>0)$ and denote $\kappa:=||E|-2|$ the distance of $E$ to the spectral edges $\pm 2$. We have proved a local version of this result for generalized Wigner matrices in the form of the following probability estimate:

that holds for any fixed positive constants $\varepsilon$ and $K$ and for any $z=E+i\eta$ such that $|E|\leq 10,\;N\eta\kappa^{3/2}\geq N^{\varepsilon}$. Note that this estimate deteriorates near the spectral edges as $\kappa\ll 1$.

In this paper we prove the following local semicircle law that provides essentially the optimal estimate uniformly in $E={\mathfrak{Re}\,}z$. We will estimate not only the deviation of $m(z)$ from $m_{sc}(z)$, but also the deviation of each diagonal matrix element of the resolvent, $G_{kk}(z)$, from $m_{sc}(z)$. Moreover, we show that the off-diagonal elements of the resolvent are small.

Our goal is to estimate the following quantities

Then there exist positive constants $A_{0}>1$, $C,c$ and $\phi<1$ depending only on $\vartheta$, on $\delta_{\pm}$ from Assumption (B) and on $C_{0}$ from Assumption (C), such that for all $L$ with

the following estimates hold for any sufficiently large $N\geq N_{0}(\vartheta,\delta_{\pm},C_{0})$:

(i) The Stieltjes transform of the empirical eigenvalue distribution of $H$ satisfies

(ii) The individual matrix elements of the Green function satisfy that

(iii) The largest eigenvalue of $H$ is bounded by $2+N^{-2/3}(\log N)^{9L}$ in the sense that

The subexponential decay condition (2.17) can be weakened if we are not aiming at error estimates faster than any power law of $N$. This can be easily carried out and we will not pursue it in this paper. We also note that the upper bound on $L$ originates from the natural requirement that ${\bf S}_{L}\neq\emptyset$.

Prior to our results in and , a central limit theorem for the semicircle law on macroscopic scale for band matrices was established by Guionnet and Anderson and Zeitouni ; a semicircle law for Gaussian band matrices was proved by Disertori, Pinson and Spencer . For a review on band matrices, see the recent article by Spencer.

The local semicircle estimates imply that the empirical counting function of the eigenvalues is close to the semicircle counting function and that the locations of the eigenvalues are close to their classical location in mean square deviation sense. Recall that $\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{N}$ are the ordered eigenvalues of $H$. We define the normalized empirical counting function by

be the distribution function of the semicircle law and recall that $\gamma_{j}=\gamma_{j,N}$ denote the classical location of the $j$-th point under the semicircle law, see (1.3).

Suppose that Assumptions (A), (B), (C) and the condition (2.17) hold. Then there exist positive constants $A_{0}>1$, $C,c$ and $\phi<1$ depending only on $\vartheta$, on $\delta_{\pm}$ from Assumption (B) and on $C_{0}$ from Assumption (C) such that for any $L$ with

for any sufficiently large $N\geq N_{0}(\vartheta,\delta_{\pm},C_{0})$.

with some small positive $\varepsilon_{0}$ was proved under the assumption that the third moment of the matrix element vanishes and all variances of the matrix elements are identical, i.e., for the standard Wigner matrices with vanishing third moment. In the same paper, it was conjectured that the factor $N^{-1/6-\varepsilon_{0}}$ on the right hand side of (2.27) should be replaced by $N^{-2/3+\varepsilon}$. Prior to the work , the estimate (2.25) away from the edges with a slightly weaker probability estimate and with the $(\log N)^{L}$ factor replaced by $N^{\delta}$ for arbitrary $\delta>0$ was proved in (see the equation before (7.8) in ). For Wigner matrices whose matrix element distributions matching the standard Gaussian random variable up to the third moment, it was proved in that $|\lambda_{j}-\gamma_{j}|\leq N^{-1+\varepsilon}$ holds in the bulk in probability (Theorem 32). More detailed behavior can be obtained if one assumes further that the fourth moment also matches the standard Gaussian random variable, see Corollary 21 of for more details. Near the edge, (2.25) with $N^{-2/3}$ replaced by $N^{-1/2}$ and the probability estimate on the right side replaced by a Gaussian type estimate was proved in .

We remark that all results in this paper are stated for both the hermitian or symmetric case, but the statements hold for quaternion self-dual random matrices as well (see, e.g., Section 3.1 of ). The proofs will be presented for the hermitian case for definiteness but with obvious modifications they are valid for the other two cases as well.

We will frequently use the notation $C$ and $c$ for generic positive constants and $N_{0}$ for the lower threshold for $N$ in this paper. We adopt the convention that, unless stated otherwise, these constants and also the implicit constants in the $O(\cdot)$ notation may depend on the basic parameters of our model, namely on $\vartheta$, $\delta_{\pm}$ and $C_{0}$. The values of these generic constants may change from line to line.

We now use Theorem 2.2 to establish the speed of convergence for local statistics of Dyson Brownian motion. In fact, we will replace the Brownian motion in the definition of Dyson Brownian motion by an Ornstein-Uhlenbeck process. We thus consider a flow of random matrices $H_{t}$ satisfying the following matrix valued stochastic differential equation

where $\beta_{t}$ is a hermitian matrix valued process whose diagonal matrix elements are standard real Brownian motions and the off-diagonal elements are independent standard complex Brownian motions; with all Brownian motions being independent. The initial condition $H_{0}$ is the original hermitian Wigner matrix. For any fixed $t\geq 0$, the distribution of $H_{t}$ coincides with that of

where $V$ is an independent GUE matrix whose matrix elements are centered Gaussian random variables with variance $1/N$. For the symmetric case, the matrix elements of $\beta_{t}$ in (2.28) are real Brownian motions and $V$ in (2.29) is a GOE matrix. It is well-known that the eigenvalues of $H_{t}$ follow a process that is also called the Dyson Brownian motion (in our case with a drift but we will still call it Dyson Brownian motion).

be the probability measure of the eigenvalues of the general $\beta$ ensemble, with $\beta\geq 1$ ($\beta=2$ for GUE, $\beta=1$ for GOE). Here $Z_{\beta}$ is the normalization factor so that $\mu$ is probability measure. In this section, we often use the notation $x_{j}$ instead of $\lambda_{j}$ for the eigenvalues to follow the notations of . Denote the distribution of the eigenvalues at time $t$ by $f_{t}({\bf x})\mu({\rm d}{\bf x})$. Then $f_{t}$ satisfies

For any $n\geq 1$ we define the $n$-point correlation functions (marginals) of the probability measure $f_{t}{\rm d}\mu$ by

With a slight abuse of notations, we will sometimes also use $\mu$ to denote the density of the measure $\mu$ with respect to the Lebesgue measure. The correlation functions of the equilibrium measure are denoted by

The main result in concerning Dyson Brownian motion, Theorem 2.1, states that the local ergodicity of Dyson Brownian motion holds for $t\geq N^{-2{\mathfrak{a}}+\delta}$ for any $\delta>0$ provided that the Assumption III (1.4) holds. In fact, the estimate on the relaxation to the local equilibrium is not restricted to Dyson Brownian motion; it applies to all flows satisfying four general assumptions, labelled as Assumption I-IV in . Instead of repeating these assumptions in their general forms, we will give only simple sufficient conditions. Assumption I requires that the probability density of the global equilibrium measure is given by a Hamiltonian of the form

holds for any fixed $E\in[2-c,2+c]$ and for any $b=b_{N}\in(0,c/2)$ that may depend on $N$. Here $p_{t,N}^{(n)}$ and $p_{\mu,N}^{(n)}$, (2.33)–(2.34), are the correlation functions of the eigenvalues of the Dyson Brownian motion flow (2.29) and those of the equilibrium measure, respectively.

Besides a weaker version of Theorem 2.3 was proved in , a similar result, with no error estimate, was obtained in for the hermitian case by using an explicit formula related to Johansson’s formula . Theorem 2.3, however, contains explicit estimates and is valid for a time range much bigger than the previous results. In particular, we mention the following three special cases:

If we choose $\delta=1-2\varepsilon^{\prime}$ and thus $t=N^{-\varepsilon^{\prime}}$, then we can choose $b\sim N^{-1}$ and the universality is valid with essentially no averaging in $E$.

If we choose the energy window of size $b\sim 1$ and the time $t=N^{-\varepsilon^{\prime}}$, then the error estimate is of order $\sim N^{-1/2}$.

If we choose $b\sim 1$, then the smallest time scale for which we can prove the universality is $t=N^{-1+\varepsilon^{\prime}}$. This scale, up to the arbitrary small exponent $\varepsilon^{\prime}$, is optimal in accordance with the time scale to local equilibrium conjectured by Dyson .

For generalized Wigner matrices with a subexponential decay, i.e. assuming (2.6) in addition to the conditions of Theorem 2.3, the universality result with no explicit error estimate holds for any time $t\geq 0$. More precisely, for any fixed $b>0$ we have

This result, with slightly stronger conditions on the distributions of the ensemble, was already proved in . Similarly to , the extension of the universality from a small positive time to zero time requires a different method, the Green function comparison theorem in our approach. The reasons of universality for zero time and time bigger than $1/N$ are very different: Theorem 2.3 shows that the local correlation functions have already reached their equilibrium under the Dyson Brownian motion flow for any time larger than $1/N$. For time smaller than $1/N$, in particular the important case $t=0$, the universality is valid because we can compare the local correlation functions at time $t=0$ with the ones generated by the flow at time $t=N^{-\varepsilon}$ with specially adjusted initial data (see, e.g., the Matching Lemma 3.4 ). The same argument as in Section 3 of can be used to prove (2.37) from (2.36). In fact, since our new version of the strong local ergodicity of Dyson Brownian motion, Theorem 2.3, holds for very short times, the two ensembles to be compared are already very close to each other. Furthermore, effective error estimates instead of a limiting statement (2.37) can also be obtained and the parameter $b$ may also be chosen $N$-dependent. For the case that $b$ is $N$-independent, the time to local equilibrium as remarked above is $N^{-1+\varepsilon}$. Hence the condition (2.6) can be replaced by the following condition: there are constants $c,\varepsilon>0$ such that

Since these extensions require only minor modifications of the current method, we will not pursue these directions in this paper.

2 Edge distribution

Recall that $\lambda_{N}$ is the largest eigenvalue of the random matrix. The probability distribution functions of $\lambda_{N}$ for the classical Gaussian ensembles are identified by Tracy and Widom to be

where the function $F_{\beta}(s)$ can be computed in terms of Painlevé equations and $\beta=1,2,4$ corresponds to the standard classical ensembles. The distribution of $\lambda_{N}$ is believed to be universal and independent of the Gaussian structure. The strong local semicircle law, Theorem 2.1, combined with a modification of the Green function comparison theorem (Theorem 6.3) implies the following version of universality of the extreme eigenvalues.

then there is an $\varepsilon>0$ and $\delta>0$ depending on $\vartheta$ in (2.17) such that for any real parameter $s$ (may depend on $N$) we have

for $N\geq N_{0}$ sufficiently large, where $N_{0}$ is independent of $s$. Analogous result holds for the smallest eigenvalue $\lambda_{1}$.

Theorem 2.4 can be extended to finite correlation functions of extreme eigenvalues. For example, we have the following extension to (2.41):

for all $k$ fixed and $N$ sufficiently large. The proof of (2.2) is similar to that of (2.41) and we will not provide details except stating the general form of the Green function comparison theorem (Theorem 6.4) needed in this case. We remark that edge universality is usually formulated in terms of joint distributions of edge eigenvalues in the form (2.2) with fixed parameters $s_{1},s_{2},\ldots$. Our result holds uniformly in these parameters, i.e., they may depend on $N$. However, the interesting regime is $|s_{j}|\leq(\log N)^{C\log\log N}$, otherwise the rigidity estimate (2.25) gives a stronger control than (2.2).

The edge universality for Wigner matrices was first proved via the moment method by Soshnikov (see also the earlier work ) for Hermitian and orthogonal ensembles with symmetric distributions to ensure that all odd moments vanish. By combining the moment method and Chebyshev polynomials, Sodin proved edge universality of band matrices and some special class of sparse matrices .

In comparison with these results, Theorem 2.4 does not imply the edge universality of band matrices or sparse matrices , but it implies in particular that, for the purpose to identify the distribution of the top eigenvalue for a generalized Wigner matrix with the subexponential decay condition, it suffices to consider generalized Wigner ensembles with Gaussian distribution. Since the distributions of the top eigenvalues of the Gaussian Wigner ensembles are given by $F_{\beta}$ (2.39), Theorem 2.4 implies the edge universality of the standard Wigner matrices under the subexponential decay assumption alone. We remark that one can use Theorem 2.2 as an input in the approach of to prove that the distributions of the top eigenvalues of the generalized hermitian Wigner ensembles with Gaussian distributions are given by $F_{2}$. Therefore the Tracy-Widom distribution also holds for any generalized hermitian Wigner ensemble with subexponential decay. But for ensembles in different symmetry classes (e.g., symmetric Wigner ensembles), there is no corresponding result to identify the distribution of the top eigenvalue with $F_{\beta}$ if the variances are allowed to vary.

Finally, we comment that the subexponential decay assumption in our approach, though can be weakened, is far from optimal, see for discussions on optimal moment assumptions. Our approach based on the local semicircle law, however, gives both the bulk and edge universality and the symmetry of the distribution of matrix elements plays no role.

Apriori bound for the strong local semicircle law

We first prove a weaker form of Theorem 2.1, and in Section 4 we will use this apriori bound to obtain the stronger form as claimed in Theorem 2.1.

for any sufficiently large $N\geq N_{0}(\theta,\delta_{\pm},C_{0})$.

We will follow the self consistent perturbation ideas initiated in . We first introduce some notations.

These quantities depend on $z$, but we mostly neglect this dependence in the notation.

The following formulas were proved in Lemma 4.2 of .

The following large deviation estimates concerning independent random variables were proved in Appendix B of .

Let $a_{i}$ ($1\leq i\leq N$) be independent complex random variables with mean zero, variance $\sigma^{2}$ and having a uniform subexponential decay

for any sufficiently large $N\geq N_{0}$, where $N_{0}=N_{0}(\vartheta)$ depends on $\vartheta$.

The following lemma (Lemma 4.2 from ) collects elementary properties of the Stieljes transform of the semicircle law. As a technical note, we use the notation $f\sim g$ for two positive functions in some domain $D$ if there is a positive universal constant $C$ such that $C^{-1}\leq f(z)/g(z)\leq C$ holds for all $z\in D$.

We have for all $z$ with ${\mathfrak{Im}\,}z>0$ that

From now on, let $z=E+i\eta$ with $|E|\leq 5$ and $0<\eta\leq 10$ and we set \kappa=\big{|}\,|E|-2\big{|}. Then we have

Following , we define the following quantities:

For the off-diagonal terms, we will use the equation (3.5). All the quantities defined so far depend on the spectral parameter $z=E+i\eta$, but we will mostly omit this fact from the notation.

The key quantities $\Lambda$, $\Lambda_{d}$ and $\Lambda_{o}$ (2.16) appearing in Theorem 3.1 will be typically small and we will prove in this section that their size is less than $(N\eta)^{-1/3}$, modulo logarithmic corrections. We thus define the exceptional (bad) event

We will always work in the complement set ${\bf B}^{c}$, i.e., we will have

We collect some basic properties of the Green function in the following elementary lemma.

2 Estimate of the exceptional events

The following lemma is a modification of Lemma 4.5 in . It improves the estimate in the sense that the control parameter depends only on $\Lambda$ but not on $\Lambda_{d}$ and $\Lambda_{o}$ (see (2.16) for definitions). Since $\Lambda$, being an average quantity, behaves better, this yields a stronger estimate.

The main reason that $\Psi$ emerges as the key controlling parameter can be seen from the following consideration. In order to estimate the off-diagonal term $G_{ij}$, we need to bound (3.5) $K_{ij}$ and thus $Z_{ij}$. By the large deviation estimate, (3.11), we have

holds with high probability. Here we have used that $\sigma_{il}^{2}\leq C_{0}/N$ from (2.4).

Applying this identity to the Green function $G=[H-z]^{-1}$, we obtain the following “Ward identity”:

where $u_{\alpha}$ and $\lambda_{\alpha}$ are the eigenvectors and eigenvalues of $H$. The term ”Ward identity” comes from quantum field theory and it represents an identity derived from a conservation law or symmetry of a system. In our case, the symmetry is generated by the global phase multiplication $e^{i\theta}$, but this connection is not important for our purpose.

where we have used the definition of $\Lambda$ in the last inequality. Notice that the control parameter $\Psi$ appears naturally in this estimate. Furthermore, it is ${\mathfrak{Im}\,}m_{sc}(z)$ which appears in the numerator, not $m_{sc}(z)$. This is the fundamental reason that we are able to obtain optimal estimate up to the edges of the spectrum. Near the edges, ${\mathfrak{Im}\,}m_{sc}(z)$ is small while $|m_{sc}(z)|$ stays near $1$.

for any sufficiently large $N\geq N_{0}(\vartheta)$.

by (2.4) and the large deviation principle for the sum of independent random variables (e.g., (3.9)). Thus

so we can work on the complement set $\Omega_{h}^{c}$. Note that

To prove the diagonal estimate (3.39), we can choose a sufficiently small $\phi>0$ (depending on $\vartheta$) and apply the large deviation bound (3.10) from Lemma 3.3 to obtain that for any fixed $i$

Since we are in the set ${\bf B}^{c}$, we have $\Lambda_{d}+\Lambda_{o}\leq(\log N)^{-2}$. Thus from (3.6) and (3.25) we have that

Similarly, for the off-diagonal estimate (3.40), for any fixed $i\neq j$, we have from (3.11) that

Using Lemma 3.5, we have $|G_{ii}|\leq C$ and $|G_{jj}^{(i)}|\leq C$ in the set ${\bf B}^{c}$. From (3.5), we can thus estimate the off-diagonal term $G_{ij}$ by

Hence we have that in the event ${\bf B}^{c}\cap\Omega_{h}^{c}$

for large enough $N$, and thus it can be absorbed into the second term. We conclude that

Inserting this bound into (3.45) and (3.47), we have proved (3.39) and (3.40). Finally, the estimate (3.36) for $\Upsilon$ and $\Lambda_{o}$ is a simple consequence of (3.50), the definition (3.20), the bound (3.27), the definition of $\Omega_{d}$ and that $\Omega^{c}\cap{\bf B}^{c}\subset\Omega_{h}^{c}$. This completes the proof of Lemma 3.6.

3 Analysis of the self-consistent equation

Now we start using the self-consistent equation (3.21). Since

the bound (3.12) allows us to expand the denominator in (3.21) as long as $\Lambda_{d}+|\Upsilon_{i}|\leq\frac{1}{2}$. In this case, using (2.12), we obtain the following equation for $v_{i}$

Recall that $B$ denotes the $N\times N$ matrix of covariances, $B=(\sigma_{ij}^{2})$. Thus we can rewrite the last equation as

We will first use this equation to estimate $v_{i}-{[v]}$, i.e. the deviation of $v_{i}$ from its average (Lemma 3.8). In the second step, we will add up (3.51) for all $i$ and obtain an equation for ${[v]}$ (Lemma 3.9). Finally, we use a dichotomy argument to estimate $\Lambda=|{[v]}|$ in Lemma 3.10.

By normalization assumption $\sum_{j}\sigma^{2}_{ij}=1$, the vector ${\bf e}=(1,1,\ldots,1)$ is the (unique) eigenvector of $B$ with eigenvalue 1. We introduce the notation

and we recall the following elementary lemma that was proven in [23, Lemma 4.8].

for some constant $C$ that only depends on $\delta_{-}$ in (2.3).

The following lemma estimates the deviation of $v_{i}$ from its average ${[v]}$:

then in the set $\Xi\cap\Omega^{c}\cap{\bf B}^{c}$ we have

for some constant $C$ depending only on $\delta_{-}$ and for sufficiently large $N$.

Applying Lemma 3.7 for $u_{i}=v_{i}-{[v]}$, we obtain

Using (3.57) to bound $\Lambda_{d}^{2}$ in (3.56), we have proved the first inequality of (3.54), the second one follows from $\Psi\leq C(\log N)^{-2}$. This completes the proof of Lemma 3.8.

In this paper we assumed that the positive constants $\delta_{\pm}$ are independent of $N$ (see (2.3)), thus $q$ is bounded and the condition (3.53) is automatically satisfied in the set ${\bf B}^{c}$, see (3.23), and therefore (3.54) can be written as

with some constant $C$ depending only on $\delta_{\pm}$.

where ${[Z]}:=N^{-1}\sum_{i=1}^{N}Z_{i}$. The implicit constants in the error terms depend only on $\delta_{\pm}$ and $C_{0}$.

By the definition of $\Upsilon_{i}$ (3.20), by the estimates (3.27) and (3.36), we have

The size of the last term of (3.51) is less than $O(\Psi^{3}+\Lambda_{d}^{3})$ which is bounded by $O(\Psi^{2}+\Lambda^{3})$ using (3.59) and (3.61). Thus we have, from (3.36) and (3.51),

Summing up $i$ and dividing by $N$, we obtain

Collecting the various error terms and using (3.61) and that $\Lambda\leq(\log N)^{-2}$ in ${\bf B}^{c}$, we obtain (3.60) from (3.65). This completes the proof of Lemma 3.9.

4 Dichotomy estimate for ΛΛ\Lambda

where we have used the simple bound $\Psi\leq 1/\log N$ and that in the set $\Omega(z)^{c}\cap{\bf B}(z)^{c}$ all $Z_{i}$, hence ${[Z]}$ can be bounded by $\Psi$ (see (3.38) and the definition of $\Omega_{d}$).

By definition of $\Psi=\Psi(z)$ (3.29), we have

where, in the last step, we have used that $\alpha(z)\sim\sqrt{\kappa+\eta}$, see (3.68), and thus ${\mathfrak{Im}\,}m_{sc}(z)\leq C\alpha(z)$ (see Lemma 3.4). We conclude from (3.66) and $|m_{sc}|\sim 1$ that

Neglecting the error term and replacing ${[v]}$ by $\Lambda$, we roughly have the equation

This inequality provides certain estimates on $\Lambda$ depending on whether $\alpha\lesssim\beta$ or not.

for sufficiently large $N$, depending on $U$. The implementation of this idea and precise estimates on $\Lambda$ is given by the following Lemma:

in the set $\Omega(z)^{c}\cap{\bf B}(z)^{c}$ and for any sufficiently large $N\geq N_{0}(\delta_{\pm},C_{0})$.

Proof. We will set $U_{0}=9(C^{*}+1)$ and let $U\geq U_{0}$ where $C^{*}$ is the constant appearing in (3.70). Depending on the relative size of $\beta$ and $\alpha$, which is determined by $z$, we will either express ${[v]}$ or ${[v]}^{2}$ from (3.70). This will correspond to the two cases in Lemma 3.10. Recalling that $|{[v]}|=\Lambda$, the last error term in (3.70) can be easily absorbed for sufficiently large $N$ and we will get a quadratic inequality for $\Lambda$.

Case 1: $\eta={\mathfrak{Im}\,}z\geq\widetilde{\eta}(U,E)$. By the definition of $\widetilde{\eta}$, in this case $\sqrt{\kappa+\eta}\geq 2U^{2}K\beta(z)$, i.e.,

by (3.68). From the choice of $U_{0}$ and $U\geq U_{0}$ we get that $\alpha\geq\beta$ and $\frac{1}{2}\alpha\geq C^{*}\beta$. Expressing ${[v]}$ from (3.70) and absorbing the $C^{*}\beta\Lambda$ term into the left hand side, we obtain

i.e. $\Lambda\geq\alpha/8$ which is larger than $\alpha/U$, or

i.e. $\Lambda\leq 8C^{*}\beta\leq U\beta$, which proves (3.73).

Case 2: $\eta={\mathfrak{Im}\,}z<\widetilde{\eta}(U,E)$. In this case $\sqrt{\kappa+\eta}\leq 2U^{2}K\beta(z)$, i.e., $\alpha(z)\leq 2U^{2}K^{2}\beta(z)$. We express ${[v]}^{2}$ from (3.70) and we get

with a constant $C^{\prime}$ depending on $U$. This quadratic inequality immediately implies that $\Lambda\leq C_{1}(U)\beta$ with some $U$-dependent constant $C_{1}(U)$. Hence we have proved Lemma 3.10.

5 Initial estimates for large η𝜂\eta

for sufficiently large $N\geq N_{0}(\vartheta,C_{0})$. Furthermore, for $\eta\geq 3$ we have the estimate

for sufficiently large $N\geq N_{0}(\vartheta,C_{0})$.

Proof. Given the estimate (3.37), for the proof of (3.79) it is sufficient to estimate the probability of $\Theta_{d}$ and $\Theta_{o}$. The estimate (3.42) still holds, but we can now bound the last term in (3.42) simply by

for any $z$, using the trivial deterministic estimate

From (3.5) and the trivial estimate (3.83), we can estimate the off-diagonal term $G_{ij}$ in the set $\Theta(z)^{c}$ by

for sufficiently large $N$. Moreover, the same argument gives

which can be inserted in the definition of $A$, (3.18), and with $N\eta\gg 1$, we get

for sufficiently large $N$. In the set $\Theta^{c}$ a similar bound holds for $h_{ii}$ and $Z_{i}$ using $\eta\leq 10$. Recalling that $\Upsilon_{i}=A_{i}+h_{ii}-Z_{i}$, and this proves (3.80).

For the proof of (3.81) it is sufficient to bound only $\Lambda_{d}$, the necessary estimate for $\Lambda_{o}$ is given in (3.84). We define $\Upsilon=\max_{i}|\Upsilon_{i}|$ and note that for $\eta\geq 3$ we have $\Upsilon\leq CN^{-1/3}$ in the set $\Theta^{c}$ by (3.80). From the self consistent equation (3.21) and the defining equation (2.12) of $m_{sc}$, we have

Using $|G_{ii}|\leq\eta^{-1}$ from (3.83) and |m_{sc}(z)|=\big{|}\int\varrho_{sc}(x)/(x-z){\rm d}x|\leq\eta^{-1}, we obtain for $\eta\geq 3$ that

By (3.12), we have $|z+m_{sc}(z)|=|m_{sc}(z)|^{-1}\geq 3$. Together with (3.86), we obtain from (3.85) that

Since the denominator satisfies $|z+m_{sc}(z)|-\Lambda_{d}\geq 3-2/3=7/3$ by $\Lambda_{d}\leq 2/3$, the estimate (3.81) follows from (3.88) and (3.80). This completes the proof of Lemma 3.11.

6 Continuity argument : conclusion of the proof of Theorem 3.1

Throughout this section fix any $U\geq U_{0}$ from Lemma 3.10 and recall the definition of $\widetilde{\eta}(U,E)$ from before this lemma. Consider first the case of $z_{1}$. Since $\eta_{1}\geq\widetilde{\eta}(U,E)$, see (3.72), we are in the first case (3.73) in Lemma 3.10. By Lemma 3.11, we have $\Lambda_{d}(z_{1})+\Lambda_{o}(z_{1})\leq CN^{-1/3}$ in the set $\Theta(z_{1})^{c}$, in particular, $\Theta(z_{1})^{c}\subset{\bf B}(z_{1})^{c}$. Moreover, by $\Lambda(z_{1})\leq CN^{-1/3}$ in the set $\Theta(z_{1})^{c}$, and (3.68), the second alternative of (3.73) cannot hold and therefore $\Lambda(z_{1})\leq U\beta(z_{1})$ in the set $\Theta(z_{1})^{c}\cap\Omega(z_{1})^{c}\cap{\bf B}(z_{1})^{c}=\Theta(z_{1})^{c}\cap\Omega(z_{1})^{c}$. Using the probability estimates (3.35) and (3.79), we have proved that

Case 2. If $\eta_{k}<\widetilde{\eta}(U,E)$, then

where $C_{1}(U)$ is given from Lemma 3.10.

Proof. We proceed by induction on $k$, the case $k=1$ has been checked in (3.89). First consider Case 1, when $k<k_{0}$ is such that $\eta_{k}\geq\widetilde{\eta}(U,E)$, i.e. (3.90) holds by the induction hypothesis. By the definition of the sequence $z_{k}$, we have

for any $i,j$. Hence $|\Lambda(z_{k})-\Lambda(z_{k+1})|\leq N^{-6}\leq\frac{1}{2}U\beta(z_{k+1})$ and thus

In other words, the estimate on $\Lambda(z_{k+1})$ is deteriorated by a factor $3/2$, but it will be gained back by the dichotomy estimate in Lemma 3.10.

Using (3.92) we also have, in $\Omega(z_{k})^{c}\cap{\bf B}(z_{k})^{c}$,

Suppose now that $k+1$ falls into the first case, $\eta_{k+1}\geq\widetilde{\eta}(U,E)$, then, from (3.75),

so by the dichotomy estimate (3.73), $\Lambda(z_{k+1})\leq\frac{3}{2}U\beta(z_{k+1})$ from (3.93) implies $\Lambda(z_{k+1})\leq U\beta(z_{k+1})$ on the set $\Omega(z_{k+1})^{c}\cap{\bf B}(z_{k+1})^{c}$. Thus (3.35), (3.93) and (3.95) imply that

by using $C^{\prime}\geq 2C$ where $C$ is the constant from (3.35). This proves (3.90), i.e. the induction step if $\eta_{k+1}$ is in the first case. If $\eta_{k+1}$ falls into the second case, i.e., $\eta_{k+1}\leq\widetilde{\eta}(U,E)$, then (3.93) gives directly the induction step, i.e. (3.91) for $k+1$.

So far we considered Case 1, i.e., we assumed that $\eta_{k}\geq\widetilde{\eta}(U,E)$. Now consider Case 2, when $\eta_{k}<\widetilde{\eta}(U,E)$ and therefore the induction hypothesis is (3.91). The argument is very similar to the previous case but $U\beta(z_{k})$ is replaced with $C_{1}(U)\beta(z_{k})$ everywhere in (3.93), (3.6) and we still obtain (3.95). Since $\eta_{k+1}<\eta_{k}\leq\widetilde{\eta}(U,E)$, we can directly refer to (3.74) to obtain the induction step, i.e. (3.91) for $k+1$. This completes the proof of Lemma 3.12.

Optimal error bound in the strong local semicircle law

We have proved Theorem 3.1 which is weaker than the main result Theorem 2.1 but it will be used as an apriori bound for the improvement. The key ingredient for the stronger result is the following lemma which shows that ${[Z]}$, the average of $Z_{i}$’s, is much smaller than the size of typical $Z_{i}$. (Notice that in the proof of Theorem 3.1, ${[Z]}$ was estimated in (3.66) by the same quantity, $\Psi$, as each individual $Z_{i}$.)

with positive constants $C,c$ depending only on $\vartheta$ in (2.17), $\delta_{\pm}$ from Assumption (B) and $C_{0}$ from Assumption (C).

With this notation, and recalling that $\Lambda_{o}(z)=\max_{i\neq j}|G_{ij}(z)|$, we then have the following lemma whose proof will be given separately in Section 7.

for any sufficiently large $N\geq N_{0}(A_{0},\psi)$.

The first version of this lemma was presented in Lemma 5.2 of where the $p$-dependence of the constant in (4.5) was not carefully tracked and the effect of the exceptional event $\Gamma$ was estimated less precisely. This was sufficient since in we applied the result for an exponent $p$ independent of $N$; as a consequence, in particular, the probability estimates for the local semicircle law were only power law and not subexponential in $N$ as here. In the current paper we allow $p$ to depend on $N$ which requires the more precise form as stated in Lemma 4.1. Furthermore, here we give a new proof that relies on a different organization of partially independent terms. The main difference is that here we separate dependences on individual matrix elements, while in we separated entire rows and columns. The new method is therefore more robust, but combinatorially more demanding.

we will apply Lemma 4.1 in the following form:

where $\Omega(z)$ is defined in Lemma 3.6.

Proof. On the right hand side of (4.5) we can split the set $\Gamma^{c}$ as

On the set \big{[}\Omega^{c}\cap{\bf B}^{c}\cap\Xi\big{]}\cup\big{[}(\Omega_{h}^{c}\setminus\Omega^{c})\cap{\bf B}^{c}\big{]}\subset{\bf B}^{c}, we estimate $\Lambda_{o}$ trivially by

Since $\Omega_{h}^{c}\setminus\Omega^{c}\subset\Omega$, we have

Suppose that for all $z\in{\bf S}_{L}$ we have

Suppose that $\Lambda(z)=o(1)$ for $\eta=10$, $|E|\leq 5$. Then in the set $\Omega^{c}\cap{\bf B}^{c}$ we have

for any $z\in{\bf S}_{L}$. Furthermore, if $\Lambda(z)\leq\alpha(z)/2$ and (4.11) hold for some $z\in{\bf S}_{L}$, then

in the set $\Omega^{c}\cap{\bf B}^{c}$, where $\alpha$ was defined in (3.67).

Proof: In the first part of the proof $z\in{\bf S}_{L}$ is fixed so we drop the $z$-dependence of various quantities. Recall (3.67), (3.68) and Lemma 3.4 for $m_{sc}$ and $\alpha\sim\sqrt{\kappa+\eta}$. From Lemma 3.9 and using (4.11), in the set $\Omega^{c}\cap{\bf B}^{c}$ we have, with $w:=[v]$, the estimate

where we have used (4.10), the definition of $\Psi$ (3.29) and that $|w|=\Lambda$. We can complete the square of the left side and obtain the inequality

where we have used that ${\mathfrak{Im}\,}m_{sc}\leq C\alpha$. We claim that in fact

also holds; indeed this is trivial if $\Lambda\leq 2\alpha$, and if $\Lambda\geq 2\alpha$ then by assumption (4.10) $\gamma\geq\Lambda\geq 2\alpha$, so $\alpha$ can be absorbed into $\gamma$ in (4.15).

with a large parameter $T$ (independent of $N$) to be specified later, and note that $\alpha_{0}\leq\gamma$ for sufficiently large $N$.

Suppose that $\Lambda\leq\alpha/2$. In this case the $w^{2}$ terms are smaller than the leading term $\alpha w$ in the left hand side of (4.14), therefore we can express $|w|=\Lambda$ and estimate it by

In the second step also used ${\mathfrak{Im}\,}m_{sc}\leq C\alpha$. In particular, the first inequality proves (4.13).

Assume now that $\Lambda\leq\alpha/2$ and $\alpha\geq\alpha_{0}$. Plugging the lower bound (4.17) on $\alpha$ into (4.18) and using the definition of $\gamma$ we obtain

Choosing $T$ as a sufficiently large constant we obtain that

under the condition that $\Lambda\leq\alpha/2$ and $\alpha\geq\alpha_{0}$. Therefore, as long as $\alpha\geq\alpha_{0}$, we have a dichotomy: either $\Lambda\geq\alpha/2$ or $\Lambda\leq\alpha/4$.

We now fix $E$ and we continuously decrease $\eta$ from $\eta=10$ to $\eta=N^{-1}(\log N)^{L}$, the lower point in ${\bf S}_{L}$. Since $\Lambda(z)\ll 1$ and $\alpha(z)$ is bounded away from zero for $\eta=10$, $|E|\leq 5$, we know that $\Lambda\leq\alpha/2$ holds for $\eta=10$. Since $\Lambda(z)$ is continuous function, by the dichotomy we have that $\Lambda\leq\alpha/4$ for all $\eta$ as long as $\alpha\geq\alpha_{0}$. In particular, $\Lambda\leq CT^{-2}\alpha_{0}$ from (4.19) which proves (4.12) in the case $\alpha\geq\alpha_{0}$.

Finally, for $\alpha\leq\alpha_{0}$, we can estimate $\Lambda$ directly via (4.16) and this proves that

from which (4.12) follows and we have thus completed the proof.

Proof of Theorem 2.1. First we explain the idea. We will prove, by an induction on the exponent $\tau$, that $\Lambda\leq(N\eta)^{-\tau}$ holds modulo logarithmic factors with a high probability. Notice that we proved this statement for $\tau=1/3$ in Theorem 3.1. Lemma 4.3 asserts that if this statement is true for some $\tau$, then it also holds for $\frac{1+\tau}{2}$ assuming a bound on $[Z]$. This bound can be obtained from Corollary 4.2 with a high probability. Repeating the induction step for $O(\log\log N)$ times, we will obtain that $\tau$ is essentially one, i.e. we get Theorem 2.1. However, we have to keep track of the increasing logarithmic factors and the deteriorating probability estimates of the exceptional sets.

From Markov’s inequality and (4.3) we obtain that

for all $z\in{\bf S}_{L}$ with a probability at least 1-\exp\big{[}-2(\log N)^{a}\big{]}. We can now apply Lemma 4.3 so that

We run the iteration until $n=2\log\log N$ so that

This proves (2.19) after renaming $2\psi/3$ to a new $\phi$. The proof of (2.21) follows from the estimate on $\Lambda$, from (3.36), (3.59) and (4.3).

Finally, to prove (2.22), we need the following Lemma.

Let $L\geq 4$ satisfy (2.18) and define the set

Then for $A_{0}$ large enough in (2.18), we have

Proof of Lemma 4.5: For $z\in{\bf U}_{L}$ we have $\kappa=N^{-2/3}(\log N)^{8L+8}\geq\eta$ and thus we have (see (3.68))

with probability larger than 1-3\exp{\big{[}-(\log N)^{2\psi L/3}\big{]}}. Here we used the probability estimate (4.3) on $P(\Omega\cup{\bf B})$ and the first bound in (3.17). Then using the values of $\kappa$ and $\eta$ in the set (4.37), we obtain

We now prove (2.22). On the set ${\bf U}_{L}$ we have

Fix $z=E+i\eta\in{\bf U}_{L}$ and define the event

it is clear that ${\mathfrak{Im}\,}m(z)\geq\frac{1}{4}(N\eta)^{-1}$ on the set $W(z)$. Using (4.41) we obtain that

Finally, we need to control the probability of a very large eigenvalue. For example, the following (not optimal) estimate was proved in, e..g, Lemma 7.2 of . We formulate the results for the largest eigenvalue $\lambda_{N}$, but analogous results hold for the smallest eigenvalue $\lambda_{1}$ as well.

Let $H$ satisfy Assumptions (A), (B), (C) and the subexponential decay condition (2.17). Then for some $\varepsilon>0$, depending on $\vartheta$, we have

Combining this lemma with (4.43) we completed the proof of (2.22).

Estimates on the location of eigenvalues

Proof of Theorem 2.2. We now translate the information on the Stieltjes transform obtained in Theorem 2.1 to prove Theorem 2.2 on the location of the eigenvalues. We will need the following Lemma 5.1 which is a special case of Lemma 6.1 proved in with the choice $A=0$. The conditions (6.1) and (6.2) stated in Lemma 6.1 of are not sufficient. Instead, the following slightly stronger assumption is necessary:

i.e., it is not sufficient to control only the imaginary part of $m^{\Delta}$. This stronger condition is needed in (6.7) of , where the imaginary part of $m$ is changed to its real part after an integration by parts. With the condition (5.1), the proof of Lemma 6.1 in remains otherwise unchanged. This immediately proves the following lemma as a special case:

We will apply this lemma with the choice that the signed measure is the difference of the empirical density and the semicircle law,

First we prove (2.26). Choose $L:=A_{0}\log\log N$, where $A_{0}$ is given in Theorem 2.1, and we define

for simplicity. By Theorem 2.1, the assumptions of Lemma 5.1 hold for the difference $m^{\Delta}=m-m_{sc}$ with $K=10$ and $U=T_{N}^{4}$ if $y\geq y_{0}:=T_{N}^{10}/N$. For $y\leq y_{0}$, set $z=x+iy$, $z_{0}=x+iy_{0}$ and estimate

Now we use the fact that the functions $y\to y{\mathfrak{Im}\,}m(x+iy)$ and $y\to y{\mathfrak{Im}\,}m_{sc}(x+iy)$ are monotone increasing for any $y>0$ since both are Stieltjes transforms of a positive measure. Therefore the integral in (5.4) can be bounded by

By definition, ${\mathfrak{Im}\,}m_{sc}(x+iy_{0})\leq|m_{sc}(x+iy_{0})|\leq C$. By the choice of $y_{0}$ and Theorem 2.1, we have

with very high probability. Together with (5.7) and (5.4), this proves that (5.2) holds for $y\leq y_{0}$ as well if $U$ is increased to $U=T_{N}^{10}$.

The application of Lemma 5.1 shows that for any $\eta\geq 1/N$

With the fact: $y\to y{\mathfrak{Im}\,}m(x+iy)$ is monotone increasing for any $y>0$, (5.8) implies a crude upper bound on the empirical density. Indeed, for any interval $I:=[x-\eta,x+\eta]$, with $\eta=1/N$, we have

This bound can be used to estimate the difference between the characteristic function of the interval $[E_{1},E_{2}]$ and the smoothed function $f_{E_{1},E_{2},\eta}$.

Since the probability to have eigenvalues outside the interval $$are extremely small, we consider only the case that all eigenvalues are inside$$. Let $E_{1}=-4$ and $E_{2}:=E\in$. Then from (5.9) and (5.10) we have that

holds for any fixed $E\in$ with an overwhelming probability. The supremium over $E$ is a standard argument for extremely small events and we omit the details. This completes the proof of (2.26) after possibly increasing $L$ (hence $A_{0}$) and decreasing $\phi$ in order to replace the $(\log N)T_{N}^{10}$ with $(\log N)^{L}$.

Now we turn to the proof of (2.25). Let $L$ as before. Fix any $1\leq j\leq N/2$ and let $E=\gamma_{j}$, $E^{\prime}=\lambda_{j}$. Setting $t_{N}=(\log N)T_{N}^{10}=(\log N)^{10L+1}$ for simplicity, from (5.11) we have

Clearly $E\leq 1$, and using (5.11) $E^{\prime}\leq 1$ also holds with an overwhelming probability. First, using (2.22) and

we know that (2.25) holds (with a possibly increased power of $\log N$ in the left hand side) if

The correct power $(\log N)^{L}$ can be restored by increasing $L$ (hence $A_{0}$) and decreasing $\phi$, as before.

Hence, we can assume that one of $E$ and $E^{\prime}$ is in the interval $[-2+t_{N}N^{-2/3},1]$. With (5.13), this assumption implies that at least one of $n_{sc}(E)$ and $n_{sc}(E^{\prime})$ is larger than $t_{N}^{3/2}/N$. Inserting this information into (5.12), we obtain that both $n_{sc}(E)$ and $n_{sc}(E^{\prime})$ are positive and

in particular, $E+2\sim E^{\prime}+2$. Using that $n_{sc}^{\prime}(x)\sim(x+2)^{1/2}$ for $-2\leq x\leq 1$, we obtain that $n_{sc}^{\prime}(E)\sim n_{sc}^{\prime}(E^{\prime})$, and in fact $n_{sc}^{\prime}(E)$ is comparable with $n_{sc}^{\prime}(E^{\prime\prime})$ for any $E^{\prime\prime}$ between $E$ and $E^{\prime}$. Then with Taylor’s expansion, we have

Since $n_{sc}^{\prime}(E)=\rho_{sc}(E)\sim\sqrt{\kappa}$ and $n_{sc}(E)\sim\kappa^{3/2}$, moreover, by $E=\gamma_{j}$ we also have $n_{sc}(E)=j/N$, we obtain from (5.12) and (5.15) that

which proves (2.25), again, after increasing $L$ and decreasing $\phi$ to achieve the claimed $(\log N)^{L}$ prefactor. This concludes the proof of Theorem 2.2.

Edge Universality

In this section, we prove the edge universality, i.e., Theorem 2.4. At the end of Section 6.1 we will give a heuristic explanation why matching the second moments is sufficient but we first need some preparation and to introduce various notations. We will consider the largest eigenvalue $\lambda_{N}$, but the same argument applies to the lowest eigenvalue $\lambda_{1}$ as well.

denote the number of eigenvalues in $[E_{1},E_{2}]$. By Theorem 2.2 (rigidity of eigenvalues), there exist positive constants $A_{0}$, $\phi$, $C$ and $c>0$, depending only on $\vartheta$, $\delta_{\pm}$ and $C_{0}$ such that with setting